add some docs

Author: jClugstor

docs/src/basics/solve.md

@@ -36,3 +36,32 @@ These are exclusively available for native `NonlinearSolve.jl` solvers.
     level of detail of the trace. (Defaults to `TraceMinimal()`)
   - `store_trace`: Must be `Val(true)` or `Val(false)`. This controls whether the trace is
     stored in the solution object. (Defaults to `Val(false)`)
+
+## Verbosity Controls
+
+NonlinearSolve.jl provides fine-grained control over diagnostic messages, warnings, and errors
+through the `verbose` keyword argument. The verbosity system allows you to control what
+information is displayed during the solve process. See [SciMLLogging.jl](https://docs.sciml.ai/SciMLLogging/dev/) for more details. 
+
+```@docs
+NonlinearVerbosity
+```
+
+### Quick Start
+
+```julia
+# Use a preset
+solve(prob, alg; verbose = NonlinearVerbosity(SciMLLogging.Standard()))
+
+# Silence all messages
+solve(prob, alg; verbose = NonlinearVerbosity(SciMLLogging.None()))
+
+# Maximum verbosity
+solve(prob, alg; verbose = NonlinearVerbosity(SciMLLogging.All()))
+
+# Custom configuration
+solve(prob, alg; verbose = NonlinearVerbosity(
+    alias_u0_immutable = SciMLLogging.WarnLevel(),
+    threshold_state = SciMLLogging.InfoLevel()
+))
+```